12.2 Binomial General Term

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📖 Main Content

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What is a Binomial Expression?

A binomial expression is any expression that has exactly two terms added or subtracted together. The word "bi" means two.

Examples of binomial expressions:

• $(a + b)$
• $(2x + 3)$
• $\left(x + \dfrac{1}{x}\right)$
• $(3 - y)$

When you raise a binomial expression to a power $n$ — like $(a + b)^n$ — and expand it (multiply it all out), you get several terms. Finding each of those terms is what this topic is all about.

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The Binomial Theorem — A Quick Recap

When you expand $(a + b)^n$, the full expansion is:

$$(a + b)^n = a^n + {}^nC_1\, a^{n-1}b + {}^nC_2\, a^{n-2}b^2 + {}^nC_3\, a^{n-3}b^3 + \cdots + b^n$$

Notice the pattern in each term:

• The power of $a$ decreases from $n$ down to $0$.
• The power of $b$ increases from $0$ up to $n$.
• The coefficient (the number in front) of each term uses a combination written as $^nC_r$.

Rather than writing out every single term (which is very slow when $n$ is large), we use a shortcut called the General Term.

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The General Term Formula

The general term of the expansion of $(a + b)^n$ is:

$\boxed{T_{r+1} = {}^nC_r \cdot a^{n-r} \cdot b^r}$

Let's break down every part of this formula so it is crystal clear: